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Toward a Quantum Theory of Tachyon fields

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Toward a Quantum Theory of Tachyon fields March 23, 2016 15:24 IJMPA S0217751X1650041X page 1 International Journal of Modern Physics A Vol. 31, No. 9 (2016) 1650041 (14 pages) c World Scientific Publishing Company DOI: 10.1142/S0217751X1650041X Toward a quantum theory of tachyon fields Charles Schwartz Department of Physics, University of California, Berkeley, California 94720, USA [email protected] Received 11 November 2015 Accepted 29 February 2016 Published 18 March 2016 We construct momentum space expansions for the wave functions that solve the Klein– Gordon and Dirac equations for tachyons, recognizing that the mass shell for such fields is very different from what we are used to for ordinary (slower than light) particles. We find that we can postulate commutation or anticommutation rules for the operators that lead to physically sensible results: causality, for tachyon fields, means that there is no connection between space–time points separated by a timelike interval. Calculating the conserved charge and four-momentum for these fields allows us to interpret the number operators for particles and antiparticles in a consistent manner; and we see that helicity plays a critical role for the spinor field. Some questions about Lorentz invariance are addressed and some remain unresolved; and we show how to handle the group representation for tachyon spinors. Keywords: Field theory; tachyons; quantization. PACS numbers: 03.30.+p, 03.50.−z, 03.65.−w, 03.70.+k, 11.10.−z 1. Introduction What of old habits do we keep and what do we change? That is always the chal- lenging question for theoretical physicists who are seeking to innovate. The idea of tachyons (faster than light particles) has been a fascination of some theorists for many decades;a but few professional colleagues nowadays grant that idea much credibility. Among the many objections have been the claims of negative energy states, anticausal behavior, and other bizarrities. My own earlier papers have shown that if we work with a conserved energy–momentum tensor, we can resolve questions about negative energy states;2 and a more recent study showed how quantum wave packet aSee the review by Recami1 which contains a list of over 600 references. 1650041-1 March 23, 2016 15:24 IJMPA S0217751X1650041X page 2 C. Schwartz considerations eliminated the main complaints about causality violation.3 That same paper showed the possibly important contributions of (classical) tachyons to cosmological studies. When it comes to quantizing tachyons, especially quantizing a tachyonic field theory, the prevalent view4 is that there is really nothing that propagates faster than light but it is just some unstable state that needs to decay; and everything conforms to ordinary concepts of causality. This paper takes just the opposite approach: assume that tachyons might exist as a particle/field system that always propagates faster than light. This poses several mathematical challenges, and here we show how to manage most of that. Sections 2 and 3 review basic ideas about the mass shell and solutions of the Klein–Gordon equation. Sections 4 and 5 look at solutions of the Dirac equation, for ordinary particles and for tachyons; and we note the suggestion of a critical role for helicity in making a distinction between particles and antiparticles. Section 6 discusses orthogonality among various plane wave states for both equations for both types of particles. In Secs. 7 and 8, we investigate models of second quantization that may lead to causal commutators or anticommutators. Causality for ordinary (slow) particles means that there is no connection between points separated by a spacelike interval in space–time, while for tachyons it means just the opposite. In Sec. 9, we identify number operators for particles and antiparticles; and in Sec. 10, we look at Lorentz invariance of our results and raise some further questions. Section 11 shows how we can handle the group representation problem for a tachyon spinor; and in Sec. 12, we summarize what has been achieved here and look forward to further work. 2. Why Canonical Quantization is Wrong Here The mathematical procedures known as “canonical quantization” have built into them certain mathematical biases that come from assumed physical behavior of ordinary (slower than light) particles and fields. I believe that some of this is shown, or at least implied, in my 1982 paper; but let me go into this here in some detail. We start with fields defined in four-dimensional space–time and we want to ask about how they propagate. For ordinary particles/fields we assume that some initial solution is contained in some finite volume of 3-space at an initial time t0 and it will also be contained in a finite volume at another time t some finite distance away. This assumption makes sense for any particle/field that can travel no faster than the speed of light; but it is not acceptable for a tachyon, which might travel at arbitrarily high speeds. For a tachyon field, we look for an alternative geometric arrangement about its propagation that makes sense with the physical understanding that we mean to describe a particle/field that can never propagate slower than the speed of light. We choose some two-dimensional surface in three-dimensional coordinate space, over all values of the time t. Our initial value assumption is that the particle (wave packet) 1650041-2 March 23, 2016 15:24 IJMPA S0217751X1650041X page 3 Toward a quantum theory of tachyon fields will pass through this surface in some finite time interval; and we can be sure that it will also pass through any other parallel surface, located a finite distance away, in a finite time interval. Note that this alternative scheme could not be used for a slower-than-light particle/field because there is the possibility for the particle to be at rest, thus it may never pass through the second surface. (Question: Are we free to use either scheme for light?) This leaves us with the question, for the tachyon field, whether this chosen surface in 3-space is an open surface or a closed surface. In most of this paper we use an open surface (z = const); but in the Appendix we chose a closed surface (r = const). I must admit that I do not have a general answer to this question. 3. Basic Issues First we review some basic properties of solutions of the free Klein–Gordon equa- tions for ordinary particles and for tachyons. They will have the space–time behavior in the form of plane waves µ ψ(x,t)= e−ipµx = eip·x−iEt , (3.1) where for ordinary particles, pµp = E2 p p = E2 p2 =+m2 . (3.2) µ − · − This equation describes the “mass-shell” and we see that it consists of a two-sheeted hyperboloid: one with E m and the other with E m. ≥ ≤− Under any proper Lorentz transformation, these positive energy and negative energy states will remain as two distinct sets of solutions. Later, they will be used to describe particles and antiparticles. However, when we consider tachyons the mass shell is different: pµp = E2 p p = E2 p2 = m2 , (3.3) µ − · − − and this is just a single surface — a hyperboloid of one sheet — in four-dimensional space. Positive and negative values of E are not separated. We now introduce spin. 4. Dirac Wave Function for an Ordinary Particle For the Dirac equation, we have the familiar free-particle solutions, µ −ipµx (E + m) p,h ψ(x)= e uh(E, p) , uh(E, p)= N | i , (4.1) hp p,h | i where p,h is a 2-component eigenfunction of the spin in the direction of p with | i eigenvalue h = 1. ± 1650041-3 March 23, 2016 15:24 IJMPA S0217751X1650041X page 4 C. Schwartz With a choice of the normalization constant, N = 1/ 2 E + m , we calculate | | the conserved four-current for this wave function as, p jµ = ψγ¯ µψ = sign(E)(E, p) , (4.2) where ψ¯ = ψ†γ0. We similarly calculate the components of the conserved energy– momentum tensorb as, i ↔ ↔ ↔ → ← T µν = ψγ¯ µ ∂νψ + ψγ¯ ν ∂µψ , ∂ = ∂ ∂ , (4.3) 4h i − T 00 = sign(E)E2 , T 03 = T 30 = sign(E)Ep, T 33 = sign(E)p2 , (4.4) where we assume that the momentum is in the 3-direction; and all other components of the tensor vanish. In both of these calculations we see that the factor sign(E) may cause some con- fusion, until we introduce annihilation and creation operators, which anticommute with one another, and let us have positive counts of particle or antiparticle number as well as energy. 5. Dirac Wave Function for a Tachyon We can start by replacing the mass m in the Dirac equation by an imaginary im; and we do the same with the plane wave solutions (4.1). We also need a different definition for the adjoint wave function, † 0 ψ¯ = ψ γ γ5 , (5.1) µ where γ5 is the unit Hermitian matrix that anticommutes with all the γ . Then, with the normalization constant N =1/√2p, we calculate: jµ = h(E, p) , T 00 = hE2 , T 03 = T 30 = hEp, T 33 = hp2 . (5.2) − − − − Here is the strong suggestion that the helicity of the tachyon wave function can serve to distinguish between particle and antiparticle 6. Orthogonalities Next, we want to consider the general superposition of plane waves, for scalar and spinor wave functions. This will require us to use an orthogonality property that is derived from a conserved current. Let me show this by defining a generalized conserved current density that involves any two solutions of the Klein–Gordon equation. ↔ → ← jµ (x)= iψ∗(x) ∂µψ (x)= iψ∗(x) ∂µ ∂µ ψ (x); ∂ jµ (x)=0 .
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